Fully extended r-spin topological field theories

完全扩展的 r-自旋拓扑场论

• 批准号: 442672550
• 负责人: Dr. Lóránt Szegedy
• 金额: --
• 依托单位: Abteilung für Mathematische Physik
• 依托单位国家: 德国
• 项目类别: WBP Fellowship
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/442672550?language=en
• 关键词: Fully; extended; spin; topological; field

The project is at the interface of pure mathematics – topological field theories (TFTs), higher category theory and algebraic geometry – and theoretical physics. I will use ideas from physics to prove theorems in mathematics.I plan to work on 2-dimensional (2d) fully extended functorial TFTs with tangential structure in the bicategorical and in the (infinity,2)-categorical setting: topological B-twist of Landau-Ginzburg (LG) models described by matrix factorisations and other topologically twisted sigma models that provide certain differential graded (dg) categories.On the way I will apply and develop new results on the LG / (fractional) Calabi-Yau (CY) correspondence, A-infinity categories and homotopy fixed points of circle actions on higher categories categories.The main objective of the project is to find interesting examples of fully extended 2d functorial TFTs on r-spin surfaces by considering different target bicategories and (infinity,2)-categories. I would like to understand the role of fractional CY varieties – related to certain categories of matrix factorisations – as targets of LG models. Another exciting source of examples should come from dg-categories related to topological twists of sigma models in physics.In the bicategorical formulation of fully extended TFTs the proven Cobordism Hypothesis, the computation of the homotopy fixed points of the 2d rotational group, the good understanding of the bicategory of matrix factorisations, and the known conditions on 2-dualisability of dg-categories can be applied.For the second part of the project I need to construct an (infinity,2)-category of matrix factorisations, I will consider algebras in infinity categories, in particular in the dg-nerve of a dg-category to serve as new targets for framed and r-spin TFTs. The project combines state-of-the-art results that became available in the past years to produce new examples of 2d fully extended r-spin TFTs. It is an important task to find such examples in dimension higher than 1 as – although classification conjectures (and theorems in the bicategorical formulation) exist – examples are almost absent in the literature.The result of this project should deliver important insights in topological filed theory and higher category theory and would make a big step towards rigorous mathematical formulation of topologically twisted sigma models in theoretical physics.

该项目是在纯数学的接口-拓扑场理论（TFT），更高的类别理论和代数几何-和理论物理。我将用物理学的思想来证明数学中的定理。我计划在双类和双类中研究具有切向结构的2维（2d）完全扩展函子TFT。（无穷大，2）-分类设置：Landau-Ginzburg（LG）模型的拓扑B-扭曲，由矩阵分解和其他拓扑扭曲的sigma模型描述，提供一定的微分分次（dg）类别。在路上我将在LG /上应用和开发新成果（分数）卡-丘（CY）对应，A-infinity范畴和更高范畴上圆作用的同伦不动点。该项目的主要目标是在r-上找到完全扩展的2d函子TFT的有趣例子。通过考虑不同的目标双范畴和（无穷大，2）-范畴来旋转曲面。我想了解分数CY品种的作用-与某些类别的矩阵因子分解-作为LG模型的目标。另一个令人兴奋的例子来源应该来自与物理学中sigma模型的拓扑扭曲相关的dg-范畴。在完全扩展的TFT的双范畴公式中，证明了协边假设，2d旋转群的同伦不动点的计算，对矩阵分解的双范畴的良好理解，并且dg-范畴的2-对偶性的已知条件可以应用。对于项目的第二部分，我需要构造一个（infinity，2）-范畴的矩阵分解，我将考虑无穷大范畴中的代数，特别是dg-范畴的dg-神经中的代数，以作为框架和r-自旋TFT的新目标。 该项目结合了过去几年的最新成果，以生产2d完全扩展的r-自旋TFT的新示例。寻找维数大于1的自相似分类图是一个重要的任务（和双范畴公式中的定理）存在--文献中几乎没有例子。这个项目的结果应该在拓扑场论和更高范畴理论中提供重要的见解，并将在理论物理中朝着拓扑扭曲西格玛模型的严格数学公式迈出一大步。